I have samples $x_i$ of lets say a random variable $X$ (euclidean distances, $X=\sqrt{Y}$, where $Y$ is the squared distance) which I computed from squared distances samples $y_i$.
I can now calculate the sample mean $\bar{x}$ and sample variance $s^2$ as:

Is it possible to somehow compute the sample mean/variance for $Y$ (squared distances)
(more effiecient since no square root computation needs to be done)
and then transform it to the sample mean and sample variance of $X$ in an easy way?

Finally, $Q = \sum_{i=1}^n Z_1^2$ has a chi-squared distribution with $n$ degrees of freedom, and $E(Q) = n$ and $V(Q) = 2n.$ Of course,
$Q$ is the squared distance from the origin in $n$-space, if each linear component is standard normal.

If this is on the right track for you, maybe look at Wikipedia on chi-squared distribution. Also the Rayleigh distribution has to do with the (Euclidean) distance between points in a plane.